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The Eccentric-Distance Sum of Some Graphs Electronic Journal of Graph Theory and Applications 5 (1) (2017), 51–62 The eccentric-distance sum of some graphs Padmapriya P., Veena Mathad Department of Studies in Mathematics University of Mysore, Manasagangotri Mysuru - 570 006, India [email protected], veena [email protected] Abstract Let G = (V; E) be a simple connected graph. The eccentric-distance sum of G is defined as X ξds(G) = [e(u) + e(v)]d(u; v), where e(u) is the eccentricity of the vertex u in G fu;v}⊆V (G) and d(u; v) is the distance between u and v. In this paper, we establish formulae to calculate the eccentric-distance sum for some graphs, namely wheel, star, broom, lollipop, double star, friendship, multi-star graph and the join of Pn−2 and P2. Keywords: eccentricity, star, path, broom, lollipop graph, double star, complete k-partite Mathematics Subject Classification : 05C10 DOI:10.5614/ejgta.2017.5.1.6 1. Introduction Let G be a simple connected graph with the vertex set V (G) and the edge set E(G). The degree of a vertex u 2 V (G) is denoted by d(u) and is the number of vertices adjacent to u. For vertices u; v 2 V (G), the distance d(u; v) is defined as the length of any shortest path connecting u and v in G and D(u) denotes the sum of distances between u and all other vertices of G. The eccentricity e(u) of a vertex u is the largest distance between u and any other vertex v of G, i.e., e(u) = maxfd(u; v); v 2 V (G)g. Let Kn, Pn, Wn, Cn and K1;n−1 denote a complete graph, path, wheel, cycle and star on n vertices, respectively [14]. Received: 4 October 2016, Revised: 15 January 2017, Accepted: 28 January 2017. 51 www.ejgta.org The eccentric-distance sum of some graphs j Padmapriya P. et al. The Wiener index is defined as the sum of all distances between unordered pairs of vertices X W (G) = d(u; v): fu;v}⊆V (G) It is considered as one of the most used topological index with high correlation with many physical and chemical indices of molecular compounds (for the recent survey on Wiener index see [4, 5]). The parameter DD(G) called the degree distance of G was introduced by Dobrynin and Kochetova [6] and Gutman [13] as a graph-theoretical descriptor for characterizing alkanes; it can be considered as a weighted version of the Wiener index and is defined as X X DD(G) = [d(u) + d(v)]d(u; v) = d(u)D(u) fu;v}⊆V (G) u2V (G) where the summation goes over all pairs of vertices in G. In fact, when G is a tree on n vertices, it has been demonstrated that Wiener index and degree distance are closely related by (see [15, 19]) DD(G) = 4W (G) − n(n − 1): Sharma, Goswami and Madan [26] introduced a distance-based molecular structure descriptor, eccentric connectivity index (ECI) defined as X ξc(G) = e(v)d(v): v2V (G) The index ξc(G) was successfully used for mathematical models of biological activities of diverse nature [9, 11, 12, 22, 25, 26]. The investigation of its mathematical properties started only recently (for a survey on eccentric connectivity index see [17]). In [8, 18, 23, 28], the extremal graphs in various class of graphs with maximal or minimal ECI are determined. In [1, 2, 7] the authors determined the closed formulae for the eccentric connectivity index of nanotubes and nanotori. Recently, a novel graph invariant for predicting biological and physical properties eccentric- distance sum was introduced by S.Gupta, M.Singh and A.K.Madan [12]. This topological index has vast potential application in structure activity/property relationships of molecules and it also displays high discriminating power with respect to both biological activities and physical properties; see[12]. The authors in [12] have shown that some structure activity and quantitative structure property studies using eccentric-distance sum were better than the corresponding values obtained using the Wiener index. It is also interesting to study the mathematical property of this topological index. The eccentric-distance sum (EDS) of G is defined as X ξds(G) = e(u)D(u): u2V (G) The eccentric-distance sum can be defined alternatively as X ξds(G) = [e(u) + e(v)]d(u; v): fu;v}⊆V (G) 52 www.ejgta.org The eccentric-distance sum of some graphs j Padmapriya P. et al. Yu, Feng and Ilic´ [27] identified the extremal unicyclic graphs of given girth having the minimal and second minimal EDS; they also characterized trees with minimal EDS among the n-vertex trees of a given diameter. Hua, Xu and Shu[16] obtained the sharp lower bound on EDS of n-vertex cacti. Ilic,´ Yu and Feng [20] studied various lower and upper bounds for the EDS in terms of other graph invariant including the Wiener index, the degree distance index, the eccentric connectivity index and so on. In this paper we establish formulae to calculate the eccentric-distance sum for some graphs, namely wheel, star, broom graph, lollipop, double star, friendship, multi-star graph and P ln graph. 2. Main Results M. J. Morgan et al. calculated the eccentric-distance sum for complete graph, cycle graph and path graph. Proposition 2.1. [3] ds 1.ξ (Kn) = n(n − 1) 8 n4 > ; if n is even > 8 ds < 2.ξ (Cn) = > n(n − 1)2(n + 1) :> ; if n is odd. 8 8 25n4 − 16n3 − 28n2 + 16n > ; if n is even > 96 ds < 3.ξ (Pn) = > 25n4 − 16n3 − 34n2 + 16n + 9 :> ; if n is odd. 96 ds Proposition 2.2. ξ (Wn) = (n − 1)(4n − 9) Proof. e(v1) = 1 and e(vi) = 2 where, i = 2; 3; : : : ; n: Then, ds X ξ (Wn) = [e(vi) + e(vj)]d(vi; vj) fvi;vj }⊆V (Wn) =(1 + 2)1(n − 1) + (2 + 2)1(2) + (2 + 2)2(n − 4) + (2 + 2)1 + (2 + 2)2(n − 4) + (2 + 2)1 + (2 + 2)2(n − 5) + ··· + (2 + 2)1 + (2 + 2)2(1) + (2 + 2)1 =(n − 1)(4n − 9): 53 www.ejgta.org The eccentric-distance sum of some graphs j Padmapriya P. et al. ds Proposition 2.3. ξ (K1;n−1) = (n − 1)(4n − 5) Proof. e(v0) = 1 and e(vi) = 2 where, i = 1; 2; 3; : : : ; n − 1: Then, ds ξ (K1;n−1) =(1 + 2)1(n − 1) + (2 + 2)2(n − 2) + (2 + 2)2(n − 3) + ··· + (2 + 2)2(1) =(n − 1)(4n − 5): Definition 2.1. [23] The broom graph Bn;d is a graph consisting of a path Pd, together with (n−d) end vertices all adjacent to the same end vertex of Pd. un−d u u4 u u3 v1 v2 v3 vd−1 vd u u u u u u u2 u Figure 1. Broom. graph Bn;d u1 u Theorem 2.1. The eccentric-distance sum of a broom graph Bn;d is d 2 d −1 3 d X X2 ξds(P ) + (n − d − 1) (d2 − 3d + 4n) + k2 + (d − k)k ; d+1 42 5 d k=1 k= 2 when d is even, d−1 2 d 3 d X X2 ξds(P ) + (n − d − 1) (d2 − 3d + 4n) + k2 + (d − k)k ; d+1 42 5 d−1 k=1 k= 2 +1 when d is odd. Proof. Let fv1; v2; : : : ; vd; u1; u2; : : : ; un−dg be the set of n vertices of the broom graph Bn;d: We consider the following cases. Case(i): d is even. d e(v1) = d; e(v2) = e(vd) = d − 1; e(v3) = e(vd−1) = d − 2; ..., e(v d ) = e(v d +2) = + 1; 2 2 2 d e(v d +1) = ; and e(u1) = e(u2) = ··· = e(un−d) = d: Then, 2 2 54 www.ejgta.org The eccentric-distance sum of some graphs j Padmapriya P. et al. ds ds ξ (Bn;d) =ξ (Pd+1) + f[d + d] d + [(d − 1) + d](d − 1) + [(d − 2) + d](d − 2) + ··· d d d d d d + + 2 + d + 2 + + 1 + d + 1 + + d 2 2 2 2 2 2 d d + + 1 + d − 1 + ··· + [(d − 2) + d] 2 + [(d − 1) + d] 1 (n − d − 1) 2 2 + (d + d) 2 [(n − d − 1) + (n − d − 2) + ··· + 2 + 1] d 2 d −1 3 d X X2 =ξds(P ) + (n − d − 1) (d2 − 3d + 4n) + k2 + (d − k)k : d+1 42 5 d k=1 k= 2 Case(ii): d is odd. e(v1) = d; e(v2) = e(vd) = d − 1; e(v3) = e(vd−1) = d − 2; ..., e(v d+1 ) = e(v d+1 ) = 2 −1 2 +2 d − 1 d − 1 + 2; e(v d+1 ) = e(v d+1 +1) = + 1; and e(u1) = e(u2) = ··· = e(un−d) = d: Then, 2 2 2 2 ds ds ξ (Bn;d) =ξ (Pd+1) + f[d + d] d + [(d − 1) + d](d − 1) + [(d − 2) + d](d − 2) + ··· d − 1 d − 1 d − 1 d − 1 + + 2 + d + 2 + + 1 + d + 1 2 2 2 2 d − 1 d − 1 d − 1 d − 1 + + 1 + d + + 2 + d − 1 + ··· 2 2 2 2 + [(d − 2) + d] 2 + [(d − 1) + d] 1g (n − d − 1) + (d + d) 2 [(n − d − 1) + (n − d − 2) + ··· + 2 + 1] d−1 2 d 3 d X X2 =ξds(P ) + (n − d − 1) (d2 − 3d + 4n) + k2 + (d − k)k : d+1 42 5 d−1 k=1 k= 2 +1 Definition 2.2.
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